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101. CJM 2007 (vol 59 pp. 1050)

Raghuram, A.
On the Restriction to $\D^* \times \D^*$ of Representations of $p$-Adic $\GL_2(\D)$
Let $\mathcal{D}$ be a division algebra over a nonarchimedean local field. Given an irreducible representation $\pi$ of $\GL_2(\mathcal{D})$, we describe its restriction to the diagonal subgroup $\mathcal{D}^* \times \mathcal{D}^*$. The description is in terms of the structure of the twisted Jacquet module of the representation $\pi$. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on restriction also shows that $\pi$ is $\mathcal{D}^* \times \mathcal{D}^*$-distinguished if and only if $\pi$ admits a Shalika model. We further prove that if $\mathcal{D}$ is a quaternion division algebra then the twisted Jacquet module is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic case.

Categories:22E50, 22E35, 11S37

102. CJM 2007 (vol 59 pp. 673)

Ash, Avner; Friedberg, Solomon
Hecke $L$-Functions and the Distribution of Totally Positive Integers
Let $K$ be a totally real number field of degree $n$. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained from geometric considerations. This result depends on unfolding an integral over a compact torus.

Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace
Categories:11M41, 11F30, , 11F55, 11H06, 11R47

103. CJM 2007 (vol 59 pp. 553)

Dasgupta, Samit
Computations of Elliptic Units for Real Quadratic Fields
Let $K$ be a real quadratic field, and $p$ a rational prime which is inert in $K$. Let $\alpha$ be a modular unit on $\Gamma_0(N)$. In an earlier joint article with Henri Darmon, we presented the definition of an element $u(\alpha, \tau) \in K_p^\times$ attached to $\alpha$ and each $\tau \in K$. We conjectured that the $p$-adic number $u(\alpha, \tau)$ lies in a specific ring class extension of $K$ depending on $\tau$, and proposed a ``Shimura reciprocity law" describing the permutation action of Galois on the set of $u(\alpha, \tau)$. This article provides computational evidence for these conjectures. We present an efficient algorithm for computing $u(\alpha, \tau)$, and implement this algorithm with the modular unit $\alpha(z) = \Delta(z)^2\Delta(4z)/\Delta(2z)^3.$ Using $p = 3, 5, 7,$ and $11$, and all real quadratic fields $K$ with discriminant $D < 500$ such that $2$ splits in $K$ and $K$ contains no unit of negative norm, we obtain results supporting our conjectures. One of the theoretical results in this paper is that a certain measure used to define $u(\alpha, \tau)$ is shown to be $\mathbf{Z}$-valued rather than only $\mathbf{Z}_p \cap \mathbf{Q}$-valued; this is an improvement over our previous result and allows for a precise definition of $u(\alpha, \tau)$, instead of only up to a root of unity.

Categories:11R37, 11R11, 11Y40

104. CJM 2007 (vol 59 pp. 503)

Chevallier, Nicolas
Cyclic Groups and the Three Distance Theorem
We give a two dimensional extension of the three distance Theorem. Let $\theta$ be in $\mathbf{R}^{2}$ and let $q$ be in $\mathbf{N}$. There exists a triangulation of $\mathbf{R}^{2}$ invariant by $\mathbf{Z}^{2}$-translations, whose set of vertices is $\mathbf{Z}^{2}+\{0,\theta,\dots,q\theta\}$, and whose number of different triangles, up to translations, is bounded above by a constant which does not depend on $\theta$ and $q$.

Categories:11J70, 11J71, 11J13

105. CJM 2007 (vol 59 pp. 372)

Maisner, Daniel; Nart, Enric
Zeta Functions of Supersingular Curves of Genus 2
We determine which isogeny classes of supersingular abelian surfaces over a finite field $k$ of characteristic $2$ contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus $2$. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to $k$-isomorphism, leading to the same zeta function.

Categories:11G20, 14G15, 11G10

106. CJM 2007 (vol 59 pp. 85)

Foster, J. H.; Serbinowska, Monika
On the Convergence of a Class of Nearly Alternating Series
Let $C$ be the class of convex sequences of real numbers. The quadratic irrational numbers can be partitioned into two types as follows. If $\alpha$ is of the first type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $\sum c_k/k$ converges. An example of a quadratic irrational of the first type is $\sqrt{2}$, and an example of the second type is $\sqrt{3}$. The analysis of this problem relies heavily on the representation of $ \alpha$ as a simple continued fraction and on properties of the sequences of partial sums $S(n)=\sum_{k=1}^n (-1)^{\lfloor k\alpha \rfloor}$ and double partial sums $T(n)=\sum_{k=1}^n S(k)$.

Keywords:Series, convergence, almost alternating, convex, continued fractions
Categories:40A05, 11A55, 11B83

107. CJM 2007 (vol 59 pp. 148)

Muić, Goran
On Certain Classes of Unitary Representations for Split Classical Groups
In this paper we prove the unitarity of duals of tempered representations supported on minimal parabolic subgroups for split classical $p$-adic groups. We also construct a family of unitary spherical representations for real and complex classical groups

Categories:22E35, 22E50, 11F70

108. CJM 2007 (vol 59 pp. 211)

Roy, Damien
On Two Exponents of Approximation Related to a Real Number and Its Square
For each real number $\xi$, let $\lambdahat_2(\xi)$ denote the supremum of all real numbers $\lambda$ such that, for each sufficiently large $X$, the inequalities $|x_0| \le X$, $|x_0\xi-x_1| \le X^{-\lambda}$ and $|x_0\xi^2-x_2| \le X^{-\lambda}$ admit a solution in integers $x_0$, $x_1$ and $x_2$ not all zero, and let $\omegahat_2(\xi)$ denote the supremum of all real numbers $\omega$ such that, for each sufficiently large $X$, the dual inequalities $|x_0+x_1\xi+x_2\xi^2| \le X^{-\omega}$, $|x_1| \le X$ and $|x_2| \le X$ admit a solution in integers $x_0$, $x_1$ and $x_2$ not all zero. Answering a question of Y.~Bugeaud and M.~Laurent, we show that the exponents $\lambdahat_2(\xi)$ where $\xi$ ranges through all real numbers with $[\bQ(\xi)\wcol\bQ]>2$ form a dense subset of the interval $[1/2, (\sqrt{5}-1)/2]$ while, for the same values of $\xi$, the dual exponents $\omegahat_2(\xi)$ form a dense subset of $[2, (\sqrt{5}+3)/2]$. Part of the proof rests on a result of V.~Jarn\'{\i}k showing that $\lambdahat_2(\xi) = 1-\omegahat_2(\xi)^{-1}$ for any real number $\xi$ with $[\bQ(\xi)\wcol\bQ]>2$.

Categories:11J13, 11J82

109. CJM 2007 (vol 59 pp. 127)

Lamzouri, Youness
Smooth Values of the Iterates of the Euler Phi-Function
Let $\phi(n)$ be the Euler phi-function, define $\phi_0(n) = n$ and $\phi_{k+1}(n)=\phi(\phi_{k}(n))$ for all $k\geq 0$. We will determine an asymptotic formula for the set of integers $n$ less than $x$ for which $\phi_k(n)$ is $y$-smooth, conditionally on a weak form of the Elliott--Halberstam conjecture.

Categories:11N37, 11B37, 34K05, 45J05

110. CJM 2006 (vol 58 pp. 1203)

Heiermann, Volker
Orbites unipotentes et pôles d'ordre maximal de la fonction $\mu $ de Harish-Chandra
Dans un travail ant\'erieur, nous avions montr\'e que l'induite parabolique (normalis\'ee) d'une repr\'esentation irr\'eductible cuspidale $\sigma $ d'un sous-groupe de Levi $M$ d'un groupe $p$-adique contient un sous-quotient de carr\'e int\'egrable, si et seulement si la fonction $\mu $ de Harish-Chandra a un p\^ole en $\sigma $ d'ordre \'egal au rang parabolique de $M$. L'objet de cet article est d'interpr\'eter ce r\'esultat en termes de fonctorialit\'e de Langlands.

Categories:11F70, 11F80, 22E50

111. CJM 2006 (vol 58 pp. 1095)

Sakellaridis, Yiannis
A Casselman--Shalika Formula for the Shalika Model of $\operatorname{GL}_n$
The Casselman--Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of $p$-adic groups that are associated to unique models (i.e., multiplicity-free induced representations). We apply this method to the case of the Shalika model of $GL_n$, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y. Hironaka, and in effect reduce many such problems to straightforward calculations on the group.

Keywords:Casselman--Shalika, periods, Shalika model, spherical functions, Gelfand pairs
Categories:22E50, 11F70, 11F85

112. CJM 2006 (vol 58 pp. 796)

Im, Bo-Hae
Mordell--Weil Groups and the Rank of Elliptic Curves over Large Fields
Let $K$ be a number field, $\overline{K}$ an algebraic closure of $K$ and $E/K$ an elliptic curve defined over $K$. In this paper, we prove that if $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each $\sigma\in \Gal(\overline{K}/K)$, the Mordell--Weil group $E(\overline{K}^{\sigma})$ of $E$ over the fixed subfield of $\overline{K}$ under $\sigma$ has infinite rank.

Category:11G05

113. CJM 2006 (vol 58 pp. 843)

Õzlük, A. E.; Snyder, C.
On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions
In a previous article, we studied the distribution of ``low-lying" zeros of the family of quad\-ratic Dirichlet $L$-functions assuming the Generalized Riemann Hypothesis for all Dirichlet $L$-functions. Even with this very strong assumption, we were limited to using weight functions whose Fourier transforms are supported in the interval $(-2,2)$. However, it is widely believed that this restriction may be removed, and this leads to what has become known as the One-Level Density Conjecture for the zeros of this family of quadratic $L$-functions. In this note, we make use of Weil's explicit formula as modified by Besenfelder to prove an analogue of this conjecture.

Category:11M26

114. CJM 2006 (vol 58 pp. 643)

Yu, Xiaoxiang
Centralizers and Twisted Centralizers: Application to Intertwining Operators
ABSTRACT The equality of the centralizer and twisted centralizer is proved based on a case-by-case analysis when the unipotent radical of a maximal parabolic subgroup is abelian. Then this result is used to determine the poles of intertwining operators.

Category:11F70

115. CJM 2006 (vol 58 pp. 580)

Greither, Cornelius; Kučera, Radan
Annihilators for the Class Group of a Cyclic Field of Prime Power Degree, II
We prove, for a field $K$ which is cyclic of odd prime power degree over the rationals, that the annihilator of the quotient of the units of $K$ by a suitable large subgroup (constructed from circular units) annihilates what we call the non-genus part of the class group. This leads to stronger annihilation results for the whole class group than a routine application of the Rubin--Thaine method would produce, since the part of the class group determined by genus theory has an obvious large annihilator which is not detected by that method; this is our reason for concentrating on the non-genus part. The present work builds on and strengthens previous work of the authors; the proofs are more conceptual now, and we are also able to construct an example which demonstrates that our results cannot be easily sharpened further.

Categories:11R33, 11R20, 11Y40

116. CJM 2006 (vol 58 pp. 419)

Snaith, Victor P.
Stark's Conjecture and New Stickelberger Phenomena
We introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraic $K$-groups of rings of integers in number fields. Our conjecture is motivic in the sense that it involves the (transcendental) Borel regulator as well as being related to $l$-adic \'{e}tale cohomology. In addition, the conjecture generalises the well-known Coates--Sinnott conjecture. For example, for a totally real extension when $r = -2, -4, -6, \dotsc$ the Coates--Sinnott conjecture merely predicts that zero annihilates $K_{-2r}$ of the ring of $S$-integers while our conjecture predicts a non-trivial annihilator. By way of supporting evidence, we prove the corresponding (conjecturally equivalent) conjecture for the Galois action on the \'{e}tale cohomology of the cyclotomic extensions of the rationals.

Categories:11G55, 11R34, 11R42, 19F27

117. CJM 2006 (vol 58 pp. 3)

Ben Saïd, Salem
The Functional Equation of Zeta Distributions Associated With Non-Euclidean Jordan Algebras
This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained.

Keywords:Zeta distributions, functional equations, Bernstein polynomials, non-Euclidean Jordan algebras
Categories:11M41, 17C20, 11S90

118. CJM 2006 (vol 58 pp. 115)

Ivorra, W.; Kraus, A.
Quelques résultats sur les équations $ax^p+by^p=cz^2$
Let $p$ be a prime number $\geq 5$ and $a,b,c$ be non zero natural numbers. Using the works of K. Ribet and A. Wiles on the modular representations, we get new results about the description of the primitive solutions of the diophantine equation $ax^p+by^p=cz^2$, in case the product of the prime divisors of $abc$ divides $2\ell$, with $\ell$ an odd prime number. For instance, under some conditions on $a,b,c$, we provide a constant $f(a,b,c)$ such that there are no such solutions if $p>f(a,b,c)$. In application, we obtain information concerning the $\Q$-rational points of hyperelliptic curves given by the equation $y^2=x^p+d$ with $d\in \Z$.

Category:11G

119. CJM 2005 (vol 57 pp. 1155)

Cojocaru, Alina Carmen; Fouvry, Etienne; Murty, M. Ram
The Square Sieve and the Lang--Trotter Conjecture
Let $E$ be an elliptic curve defined over $\Q$ and without complex multiplication. Let $K$ be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes $p \leq x$ for which $\Q(\pi_p) = K$, where $\pi_p$ denotes the Frobenius endomorphism of $E$ at $p$. More precisely, under a generalized Riemann hypothesis we show that this number is $O_{E}(x^{\slfrac{17}{18}}\log x)$, and unconditionally we show that this number is $O_{E, K}\bigl(\frac{x(\log \log x)^{\slfrac{13}{12}}} {(\log x)^{\slfrac{25}{24}}}\bigr)$. We also prove that the number of imaginary quadratic fields $K$, with $-\disc K \leq x$ and of the form $K = \Q(\pi_p)$, is $\gg_E\log\log\log x$ for $x\geq x_0(E)$. These results represent progress towards a 1976 Lang--Trotter conjecture.

Keywords:Elliptic curves modulo $p$; Lang--Trotter conjecture;, applications of sieve methods
Categories:11G05, 11N36, 11R45

120. CJM 2005 (vol 57 pp. 1215)

Khare, Chandrashekhar
Reciprocity Law for Compatible Systems of Abelian $\bmod p$ Galois Representations
The main result of the paper is a {\em reciprocity law} which proves that compatible systems of semisimple, abelian mod $p$ representations (of arbitrary dimension) of absolute Galois groups of number fields, arise from Hecke characters. In the last section analogs for Galois groups of function fields of these results are explored, and a question is raised whose answer seems to require developments in transcendence theory in characteristic $p$.

Category:11F80

121. CJM 2005 (vol 57 pp. 1102)

Weston, Tom
Power Residues of Fourier Coefficients of Modular Forms
Let $\rho \colon G_{\Q} \to \GL_{n}(\Ql)$ be a motivic $\ell$-adic Galois representation. For fixed $m > 1$ we initiate an investigation of the density of the set of primes $p$ such that the trace of the image of an arithmetic Frobenius at $p$ under $\rho$ is an $m$-th power residue modulo $p$. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals $1/m$ whenever the image of $\rho$ is open. We further conjecture that for such $\rho$ the set of these primes $p$ is independent of any set defined by Cebatorev-style Galois-theoretic conditions (in an appropriate sense). We then compute these densities for certain $m$ in the complementary case of modular forms of CM-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem (which does apply in the CM case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at $p$ in abelian extensions of imaginary quadratic fields unramified away from $p$.

Categories:11F30, 11G15, 11A15

122. CJM 2005 (vol 57 pp. 1080)

Pritsker, Igor E.
The Gelfond--Schnirelman Method in Prime Number Theory
The original Gelfond--Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on $[0,1]$ to give a Chebyshev-type lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's $\psi$-function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support.

Keywords:distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials
Categories:11N05, 31A15, 11C08

123. CJM 2005 (vol 57 pp. 812)

Trifković, Mak
On the Vanishing of $\mu$-Invariants of Elliptic Curves over $\qq$
Let $E_{/\qq}$ be an elliptic curve with good ordinary reduction at a prime $p>2$. It has a well-defined Iwasawa $\mu$-invariant $\mu(E)_p$ which encodes part of the information about the growth of the Selmer group $\sel E{K_n}$ as $K_n$ ranges over the subfields of the cyclotomic $\zzp$-extension $K_\infty/\qq$. Ralph Greenberg has conjectured that any such $E$ is isogenous to a curve $E'$ with $\mu(E')_p=0$. In this paper we prove Greenberg's conjecture for infinitely many curves $E$ with a rational $p$-torsion point, $p=3$ or $5$, no two of our examples having isomorphic $p$-torsion. The core of our strategy is a partial explicit evaluation of the global duality pairing for finite flat group schemes over rings of integers.

Category:11R23

124. CJM 2005 (vol 57 pp. 494)

Friedlander, John B.; Iwaniec, Henryk
Summation Formulae for Coefficients of $L$-functions
With applications in mind we establish a summation formula for the coefficients of a general Dirichlet series satisfying a suitable functional equation. Among a number of consequences we derive a generalization of an elegant divisor sum bound due to F.~V. Atkinson.

Categories:11M06, 11M41

125. CJM 2005 (vol 57 pp. 535)

Kim, Henry H.
On Local $L$-Functions and Normalized Intertwining Operators
In this paper we make explicit all $L$-functions in the Langlands--Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local $L$-functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for $\re(s)\geq 1/2$ in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrum and determining poles of automorphic $L$-functions.

Categories:11F70, 22E55
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